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If both these sentences are past tense, then shouldn't they both use "met"?

Have we ever met?

Did we ever meet?

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Have we ever met?

In this sentence, you don't actually have the past tense. You have the Present Perfect here. The Present Perfect is formed in the following way: the verb 'to have' in the Present Simple—which is either 'have' or 'has'—plus the 3rd form of the verb (Past Participle). So the word 'met' in 'Have we ever met?' is a Past Participle form and not past tense.

Did we ever meet?

Here we have another English tense, the Past Simple. But please notice that we have an interrogative sentence. More exactly, we have a yes/no question. In English, interrogative sentences in the Past Simple (excluding questions where the wh- word is the subject) are constructed using the auxiliary verb 'do' in the Past tense (that is, 'did') plus the infinitive of the main verb without 'to'. The subject is usually placed between the auxiliary and the main verb. You cannot use the 2nd form, the Past Simple of the main verb. Thus, you use 'meet', which is infinitive without 'to', and not 'met'.

A few words about usage of these sentences. Personally I consider the first one is more correct. We have the word 'ever', so I would use Present Perfect here ('Have we ever met?'). I'm not comfortable with the Past Simple. But maybe in American English you could use it as well.

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(edit) It's not gramatically appropriate to use ever with past simple, unless in some colloquial forms (typically American).

Therefore, the only correct way of saying this sentence is:

Have we ever met?

The only situation I was able to find where ever is used with past simple was this (link to the source - Meaning of “ever” in the English Dictionary):

As ever, I was the last to find out.

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  • Did you ever is appropriate as well as "have you ever". – user178049 Mar 30 '17 at 9:49
  • Actually, it's only acceptable to use it in speech. Edited that in. – critical_mass Apr 17 '17 at 18:25